We study homological mirror symmetry for $(\mathbb{ P}^2, \Omega)$ viewed as an object of birational geometry, with $\Omega$ the standard meromorphic volume form. First, we construct universal objects on the two sides of mirror symmetry, focusing on the exact symplectic setting: a smooth complex scheme $U_\mathrm{ univ}$ and a Weinstein manifold $M_\mathrm{ univ}$, both of infinite type; and we prove homological mirror symmetry for them. Second, we consider autoequivalences. We prove that automorphisms of $U_\mathrm{ univ}$ are given by a natural discrete subgroup of $\operatorname{ Bir} (\mathbb{ P}^2, \pm \Omega)$; and that all of these automorphisms are mirror to symplectomorphisms of $M_\mathrm{ univ}$. We conclude with some applications.
Categories: math.SG math.AG